where we use the symmetry property of Rademacher variables \(\varepsilon _i\) . By Eq. (4.20) in [1]}, we can further bound this term by
\(2\mathbb {E}\sup _{f\in \mathcal {F},\Vert \mathbf {w}\Vert _q=R}g*{\sum _{i=1}^m \varepsilon _i\sigma *{\mathbf {w}^\mathrm {T}f(\mathbf {x}_i)}} &\le 2\mathbb {E}\sup _{f\in \mathcal {F},\Vert \mathbf {w}\Vert _q=R}g*{L*{\mathbf {w}}_q*{\sum _{i=1}^m \varepsilon _if(\mathbf {x}_i)}_p} \\&=2\mathbb {E}\sup _{f\in \mathcal {F}}g*{LR*{\sum _{i=1}^m \varepsilon _if(\mathbf {x}_i)}_p},\) 
